underlying Principles of Leg Motion  in skating:  physics + biomechanics
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### what's here

[ under construction ]

## intro

?? [ more to be added ]

## ways to create propulsive work

For the leg-push of skating, propulsive work is created in a couple of ways:

(a) Most obviously by moving some part(s) of the body roughly opposite to the intended propulsive pushing direction of the foot against the ground. This "pushing direction" is usually along the surface of the ground, at a right angle to the aiming direction of the skate or ski, pointed partly sideways and partly backwards. So to move a body part opposite to direction that means sideways the other way and/or forward.

Sometimes the parts getting moved away are small (e.g. the forearms, when making an arm-swing move), and other times large, sometimes almost the whole body (e.g. all body parts above the ankle, when making a toe-push move).

Most real push moves are mixtures of side-side, front-back, up-down, and usually the proportions of the mixture change during the move. So a move which is primarily upward might have a small propulsive contribution because it has some side-side mixed in with it.

(b) Slowing or stopping a body part is sometimes propulsive. If a body part is already moving roughly toward the pushing direction, then stopping that part will add force in the pushing direction (by Newton's Third Law about action + reaction).

The positive propulsive effect from stopping the motion of a body part can be just as real and large as from starting a motion.

The negative propulsive effect from stopping the motion of a body part could be also be just as real and large as the positive starting -- which is a problem which must be carefully managed in the cycle sequence of skating motions, because every body-part move that gets started must also be stopped sometime during the cycle. Have to be clever to avoid having the propulsion effects from the stopping and the starting exactly cancel each other.

(c) Sometimes can make a move which stores energy which can be converted into a propulsive work of type (a) or (b) later.

The classic case is to lift parts of the body early when the foot is underneath the hip, then let those parts fall later when the foot is out to the side, to add force to the push through the foot outward and backward. It's really only propulsive when the overall average vertical position ("center of mass") of the whole body moves upward -- not when one part moves up at the cost of another part moving down.

Another important case is the "side-weight-shift" energy which starts the mass of the body moving away from the push but later is found to be moving toward the next push, so then it can add work of type (b) to that future push. In this case the same move both adds work to the current push and stores energy to be used in the next push.

"direct" versus "reactive force" / "inertial" moves?

Question: Are some propulsive moves "direct" and others "reactive force" or "inertial force"?

Answer: There's no clear distinction like that which is helpful. Virtually all "currently propulsive" moves (type (a) and (b)) in skating have some sideways component -- and for the moves making the larger contributions to power, the sideways component is a rather substantial proportion of the force applied. The sideways component works by pushing "against" the inertia of the mass of some body part(s) and thus generating a reaction force which gets added to the total force through the skater's foot to the ground. So all the important currently propulsive moves in skating are substantially "reactive force".

The joint for some moves is much closer to the foot (e.g. knee-extension) than for others (e.g. arm-swing). The closer moves tend to move a larger mass (e.g. everything hips and higher) away from the pushing-direction, and tend to have less power-transmission losses. Maybe that last is what "direct" could mean, but it's a continuous scale, not a clear qualitatitive distinction.

Whether the joint is closer or farther from the foot, the skater still needs to worry about timing its move with aiming-angle changes and leg-switches, so that starting and stopping do not cancel each other, and design the recovery move so it does not cancel the primary push.

A move is currently propulsive if it applies a force with a component toward the current intended pushing-direction. It is true that one component the propulsive force through the foot pushes "against" the overall frictional and air resistance against the forward motion of the skater, while the other component pushes "against" the inertia of the skater's side-to-side motion. But the proportion between those two components is determined by the aiming-angle of the skate or ski, not by which body-part moves are being performed.

Physics more formally:

Really type (a) and type (b) are pretty much the same thing in the formal physics: Applying a force to an object to produce acceleration and a reactive force. Type (b) is just a negative acceleration.

Rotational motion: Though it's easier to think about forces and linear motion, because of how bones and joints work, it's usually more accurate to think of most cases where a muscle moves a body part as applying a torque through a joint to produce rotational motion.

n = unit vector of the intended propulsive pushing direction.

rj = position of the joint through which this push is being applied, relative to the foot's contact with the ground.

τj = torque applied through this joint by this simple push.

f = force from push through that joint which gets applied at foot-ground-connection position (ignoring losses in transmission through the skater's body structures).

Then

f  =  (rj × τj) / |rj|2  where rj × τj is the vector cross-product, and |rj| is the magnitude of rj.

"Currently Propulsive" push: A simple push is "currently propulsive" if the force through the foot has a significantly positive component in the intended propulsive pushing direction; i.e.

f n > 0   or   (rj × τj) n > 0

## how much propulsive work is created

Here we address how positive and negative work is quantified, how the positive at one time in the stroke-cycle tends to get cancelled out negative at another time, and the clever tricks use in skating to create an overall net positive.

### Work definition

In physics, "work" is a very specific concept with a very specific quantification. The basic concept is that "work" is the increase of a kind of "energy" which is useful for some desired purpose (or the prevention of a decrease in useful "energy"). In this analysis of skating, the desired purpose is to maintain speed of the skater's body moving forward against resistive forces such as friction and air resistance.

"Energy" is also a very specific carefully-defined concept, with a very specific quantification -- the same as for "work". But "energy" is a quantification of the state of a system at a point in time, while "work" is a quantification of the difference ot the system state between two times. The skater's body moving at speed has a useful energy, which fits the rigorous definition of "energy" in physics, a kind called "kinetic energy".

The kind of "work" being done on this energy is a bit tricky. It is mainly not an increase in this "kinetic energy", but rather the difference between what the kinetic energy level was at one point in time versus what it would have decreased to at a later time, if this propulsive work had not been done, if the resistive forces of friction and air resistance had operated without opposition.

Usually an easier way to calculate work for linear motion is to use this formula for the Work performed between two points in time:

Work = (Force externally applied) * (Distance thru which force is applied)

This avoids direct reference to energy levels, but it can be proven that the resulting quantification is the equal. This simple formula assumes force is constant. If it varies, then instead need to express the formula as an "integral", which is complicated for non-physicists, but a straightforward extension of the definition.

Physics more formally: Since the push through a skeletal joint is rotational, it is more accurate to calculate Work = Torque * Angular-difference. The practical implications for skating technique are similar (except in special cases where they're different -- physicists can tell which cases are "special"). Anyway when the push moves get applied at the key interface between the skater's foot and the ground, it's the linear-motion concepts which are more helpful.

### Limited Range-of-Motion

This calculation of Work has some implications for moves of body parts which are restricted to a limited range of motion -- which for skating is all of the body parts. Rockets avoid this restriction by propelling the mass of their fuel backward infinitely into space. But human skaters feel a need to keep all their body parts and equipment together with them for many repetitions of the stroke-cycle.

So sometime after start of a move, that body part must also stop relative to the skater's center-of-mass. That stopping takes up time and uses part of the limited range-of-motion distance for that body part. But this required stopping does not generate propulsive work in the way that the starting did (indeed normally it opposes the starting, but that's not the immediate point), so the propulsive Work done is normally less than could be gained if the skater could use the whole range-of-motion positively (and just throw away the body part).

If the magnitude of the stopping force is constant and the same as the starting force (but in the opposite direction), then propulsive work from starting the body part move is the same as the kinetic energy at maximum velocity which occurs at the half-way point:

Work  =  ½ * (Force) * (Range-of-Motion distance).

Limited range-of-motion also has some implications for:

maximum Speed of body part =

= square-root { (Range-of-Motion) * (Force)    (Mass-of-Body-Part) }

Time duration of move =

= square-root { (Range-of-Motion) * (Mass-of-Body-Part)    (Force) }

These formulas for Speed and Time have implications for the "rhythm" of the moves. Note that a move of a body part with smaller mass tends to require higher speed and quicker time, which can cause problems -- see below under "move body part with small mass".

### self-cancellation problem 1:  Joint motions push body parts apart

Although we think of propulsive push as away from the intended pushing-direction, any push move through a skeletal joint actually pushes some body part(s) toward the pushing-direction.

Because any human joint is between some body parts, and a push move through the joint tends to push the parts on each side in different and somewhat opposing directions. In the case of skating, some other part(s) are on the side of the joint away from the foot-ground-connection (and tend to get pushed away from the pushing-direction, positive for Work), but some other part(s) are between the pushing joint and the foot-ground-connection (and tend to get pushed toward the pushing-direction, which could be negative for Work). So the positive push-move generates its own (potential) negative.

Q: How prevent (or at least reduce) the potential negative Work?

A: Do not allow the joints and bones and other bodily structures which are toward the foot-ground-cancellation to actually move through any significant distance. The "Force" part of the "Work" calculation is required in this case, but the "Distance" is not. If the skater can hold the "Distance" to (near) zero, then the negative Work will be (near) zero. This requires holding some bones and joints and structures in a stable configuration, not collapsing.

### self-cancellation problem 2:  Starting versus Stopping

A problem with generating forces by moving body parts is that starting and stopping tend to produce forces in opposite directions, by Newton's Third Law.

Note that sometimes it's the stopping that's positive and the starting which is negative for propulsion. Other times the roles are reversed.

We'll consider several cases:

• moving body part(s) primarily side-to-side. Here there's lots of starting and stopping, because of course the skater's body doesn't move very far in one sideways direction before it's time to move it back in the opposite direction.

So the self-cancellation problem is big in this case, but the solution is straightforward: Change the direction of the intended propulsive pushing direction in between the starting and the stopping, by changing the aim-angle of the ski or skate, from pointing toward one side of the overall forward motion to pointing toward the other side. Since the magnitude and direction of the force transmitted to the ground depends heavily on the aim-angle (see details below), this is way to get positive propulsive work out of both the starting and stopping of a sideways motion.

So the key is to carefully coordinate the timing of starting and stopping of sideways moves around changes in the aim-angle of the skate or ski.

In normal single-push stroking, this switch in directions is easily accomplished by switching with foot is on the ground. In double-push stroking it's trickier, done with a pivot of the same foot in the Aim-switch phase.

It does not matter whether the body parts being pushed sideways are big or small, the principle is the same. Even when nearly the entire mass of the body is being pushed, as by the ankle-extension move at the end of Phase 3, much of the benefit of that move would be cancelled if the skater somehow kept weight on the same skate or ski, pointing toward the same side.

• moving body part(s) mainly up-and-down. Here there's no contribution to currently propulsive work, so self-cancellation of starting and stopping is not a concern. As long as they overall raising of the center-of-mass is accomplished in time for when its needed in the next repetition of the stroke-cycle, the details of timing and acceleration/deceleration are usually not significant for propulsive performance results.

• moving large-mass, large-cross-section-area sets of body parts forward and backward. If the skater is succeeding in maintaining a steady cruising speed, starting-versus-stopping cancellation is not as much of a problem as in side-side motion.

Because sets of body parts which include a large percentage of the skater's body usually cannot be starting and stopping during the stroke-cycle -- otherwise the skater's center-of-mass would be coming close to starting and stopping, and the skater would not be said to be cruising steadily.

For moves of large portions of the skater's body forward-backward, the push is mainly "against" the forward-motion-resistive forces of air resistance, friction, hill-slope -- not "against" inertia. Those resistive forces are generally fairly steady, so the accelerations and decelerations of the skater's speed are not a large percentage of the average speed -- completely unlike the big swings and reversals of side-weight-shift moves.

• moving a small-mass, small-cross-section body part forward-and-backward is trickier to manage.

Two examples are extra backward-then-forward move in leg-recovery and set-down, and backward-forward arm-swing. To some extent a backward-to-forward move pushes "against" the forward-motion-resistive forces (instead of "against" inertia), but trying to take advantage of this concept runs into a different self-cancellation problem of Recovery move versus primary (see below).

The problem with forward-backward moves is that the trick with side-weight-shift moves, of just switching sides of the intended pushing-direction, doesn't work for forward-backward moves, because a forward-backward move does not have a positive or negative "side".

What makes a difference for transmission of forward-backward move is not the direction or sign of the aim-angle, but rather the magnitude of the aim-angle α.

Whenever α is in the range 0° α 45°, larger α yields greater transmission of forward-backward force into both currently-propulsive and future-propulsive work -- see the "x x" and "x y" columns in the table of transmission and conversion ratios, further below.

So the trick is change the magnitude of the aim-angle in between the positive part of the move and the negative part. If the angle is larger during the positive part and smaller in the negative part, then the total effect will be net positive for propulsive work.

Example -- extra backward-then-forward move in leg-recovery and set-down: The trick in this case is to set-down in Phase 0 with the aim-angle smaller and the skate or ski pointed closer to straight in the overall forward motion direction. Continue that smaller angle into the early part of Phase 1. Then pivot the skate to aim more outward, so the aim-angle is larger by the the leg-push is finishing in Phase 3. The timing coordination is roughly like this: Reach the farthest backward position in the leg-recovery move during Phase 3 of the other leg's push, and start the set-down move also during Phase 3 of the other leg, so the positive reactive force from accelerating the mass of the recovering leg is transmitted well into the push of the other leg. The deceleration and stopping of the forward motion then occur early in Phase 1 of this leg's next push, but the negative reactive force is not transmitted well into the ground, because the aim-angle is smaller in Phase 1. Result: the carefully timed difference in aim-angle magnitude and transmission effectiveness yields a net positive in propulsive work.

[ Running: Interesting that a slightly different trick is available to make the backward-to-forward leg-recovery and set-down move into a net positive for Running. The recovering leg starts its forward set-down move while the other foot is on the ground, so the positive reactive force is transmitted effectively. But much of the deceleration is accomplished while both feet are up in the air. So some of the negative reactive force does not get transmitted into the ground at all, so there's an overall net positive for propulsion. A similar timing trick works for getting net positive work from forward-backward arm-swing in Running.

[ Walking: but this timing trick is not available with Walking, because at least one foot is always on the ground, so all reactive forces both positive and negative are transmitted fully. Arm-swing is still used for balance in fast walking, but experienced walkers climbing up a hill generally do not swing their arms -- just let them hang down at the side -- because arm-swing doesn't help propulsion in walking. Nor does extra backward-forward motion in leg-recovery, so experienced Walkers do not do that either. ]

### self-cancellation problem 3:  Recovery move versus Primary

Recovery move problem for type (a) and (b):

* must subtract the negative work of the reactive force and the recovery move through the whole stroke-cycle.

* "locking in": above hips versus hips and below.

?? [ more to be added ]

### problem 4: Move of body part with small mass

* Muscles not good at generating high force + high velocity at the same time.

* Time duration of move gets small, tricky to coordinate timing of start versus stop with changing the intended-pushing-direction. (especially if the rhythm of the pushing-direction changes is driven by moving larger-mass parts).

?? [ more to be added ]

## Force gets "totaled" at foot-ground-connection

What matters for propulsion is not the force or torque from any particular move, but what all of them together combine into as a force applied at the position of the foot-ground-connection.

So it is not necessary to align the force direction of each move with the intended pushing-direction.

As long as a given move has some substantial component in the intended pushing-direction, that move can add propulsive work. Often the components which are off from the pushing-direction are accidentally cancelled by off-pushing-direction components of other moves, or can be cancelled deliberately by non-propulsive control moves or by modifications of certain other propulsive moves.

"All of them" includes not just forces from muscle moves, but also forces of resistance and gravity.

So the total force through the foot-ground-connection ("fgc") is

ffgc  =

forces from all muscle moves on body parts

+ weight of the skater's body

+ air resistance

+ gliding friction

+ hill slope resistance

Here's more specifics on each . . .

#### muscle push moves of body parts

The force applied by a muscle move at the foot-ground-connection ("fgc") is dependent on

• the position and orientation of the joint through which the push is made (symbol rj)

• the orientation of the joint and the magnitude of the push (described by the symbol Tj)

• how much force is lost in transmission, by being absorbed by body structures or the unintended collapsing of other joints "along the way" in between the joint and fgc.

Formula:

The force through the foot-ground-connection from the push through joint j is:

f  =  (rj × τj) / |rj|2

for more detail and explanation, see above under ways to create propulsive work.

"Gearing"

This formula implies that the further away the joint is from fgc (larger magnitude of |rj|, the smaller the force (but the higher the linear speed). So the push-moves through joints further away from the foot tend to be better for "high gear" situations (higher speed, lower required force intensity), while the joints closer to the foot position rfgc tend to be better for "low gear" situations (higher required force intensitey, lower speed).

On the other hand, you use what you got. Even if a move is not the more effective "gear" for your situation, if it helps more than it hurts, makes sense to use it. For example, arm-swing is a "high gear" kind of move, but it's helpful for climbing up a steep hill.

Anyway it's more complicated than that, because the main "gearing" control in skating is the aim-angle of the skate or ski. By reducing the aim-angle, the relative speed at fgc of the foot pushing out from the skater's body might be slower, even the absolute speed over the ground is higher.

#### weight + gravity

Gravity always has a downward force based on the total mass of the skater's body. Whether it changes current propulsion work depends on the positional configuration of the skater's body -- and not on upward or downward accelerations (parallel to the z direction).

If the skater is in balance with center-of-mass directly vertically above the foot-ground-connection, then there is no propulsive force resulting from gravity.

But if the skater's body mass is out of balance, then there is a non-vertical component of propulsive force into the foot-ground-connection, based on how far the angle from the center-of-mass to foot-ground-connection is from straight vertical. The amount of propulsive work added in a period of time depends on how far the center-of-mass drops during that period (which depends partly on previous moves in the stroke-cycle). As the foot moves out further away from underneath the hip, the CoM-foot-slant angle increases, and it's harder for the leg to support the skater's upper body, so it's normal for the skater's overall body and center-of-mass to fall during the later Phase 3 of the leg-push.

Of course in order for the stroke-cycle to be repeatable, the center-of-mass must get raised up again, and that takes real work of type (c), usually done in Phase 1 just after set-down. It's real work being done, but it's not currently propulsive.

Actually it has a negative effect on current propulsive work since moving the center-of-mass upward is also somewhat away from the foot-ground-connection, but not as a result of force in the pushing-direction, so it somewhat "softens" the force in the pushing direction. But it's usually worth accepting that cost in order to make such an effective use of the strong knee-extension muscles.

Formulas:

The CoM-foot-slant angle β is calculated by ignoring any component of the position of the Center-of-Mass relative to foot-ground-connection which is parallel to the aiming-angle of the skate or ski, the angle between that and vertical:

β  =  arccos { (r z  square-root[ (r n)2 + (r z)2 ] }

Then the magnitude of currently propulsive component of force, in the pushing-direction is:

|fn|  =  M g cos β sin β  =  ½ M g sin 2β

The configuration for maximum propulsive force is at β = 45°, at which point the force is half the skater's body weight. But the skater's body is dropping fast by then, so that force is only brief and temporary until the skater sets the other foot down to stop the falling.

#### glidingfriction

For most of our analysis of skating we assume that gliding friction is negligible in comparison with air resistance, as on ice or very smooth pavement or hard snow.

If there is significant gliding friction in the aiming-direction, it makes the analysis (and practical implications) more complicated, but the analysis is already plenty complicated and interesting without it.

Friction is proportional mainly to "coefficient of gliding friction" and to the component of force at the foot-ground-connection which is perpendicular to the surface of the ground.

Since this friction is directed opposite to the aiming-direction, it usually has a significant sideways component. This means that if we get precise in situations with significant gliding friction, the sideways component of propulsive force is partly "against" resistive force, and partly currently propulsive. Although it's still mainly "against" inertial force and substantially future-propulsive.

#### air resistance

Air resistance is the main force opposing forward motion of the skater on flat ground. It is roughly proportional to the square of the skater's velocity, so it is very important for determining the skater's speed.

Magnitude and direction of air resistance force is largely based on current forward velocity and current sideways velocity of skater's center-of-mass, and on cross-section area, turbulence, and specific-body-part-velocity deviations due to specific body-part moves

On flat terrain, the main desired purpose of propulsive work is in counter-acting the slowing effect of air resistance. On flat terrain, a major consideration limiting use of certain moves and positions is that they increase the amount of air resistance force.

The direction of air resistance is roughly opposite to the direction of the skater's center-of-mass. Since the center-of-mass moves somewhat side-to-side, the main resistive force also often has a sideways component. This means that if we get precise, the sideways component of propulsive force is partly "against" resistive force, and partly currently propulsive. Although it's still mainly "against" inertial force and substantially future-propulsive.

#### slope of hill

Modifying the analysis and formulas for the situation of skating up a hill makes everything more complicated in the details.

First the positional coordinate frame needs to be re-defined:

Because the "upward" direction given by the unit-vector z is no longer straight vertical. It is defined as perpendicular to the ground surface, but the ground is no longer horizontal. On a hill slope, we can define the upward direction as:

z = ( (cos γ) g (sin γ) (g × y) )    |g|

where

y is the sideways direction which remains pure horizontal, unchanged by the slope, and

(g × y  |g| is the vector-cross-product which yields the pure-horizontal forward unit vector which is x projected onto the pure-horizontal plane (which is perpendicular to g).

So then we get the expected result:  cos γ z g    |g|

and the "forward-motion" direction given by the unit-vector x is not longer pure horizontal, but now includes a vertical component. On a hill slope, we can define the forward direction as:

x = ( (cos γ) (g × y) - (sin γ) (g) )    |g|

So then we get the expected result:  sin γ x g    |g|

The slope of the hill also has a big impact on at least one key force:

Resistive force opposing forward motion =

= (air resistance) + (gliding friction) + g sin γ

?? [ more to be added ]

## how Force gets "de-composed" at fgc

The total net force into the foot-ground-connection ("fgc") is split into directional components, in two conceptual stages:

### (1) split into three dimensions

The total force through the foot, ffgc is split into three orthogonal directions: upward-downward, longitudinal, and current propulsion, so ffgc = fn + fa + fz:

#### up-down fz component

fz up-down component goes downward parallel to z, and perpendicular to the ground surface plane.

Formula: fz = (ffgc z) z.

To calculate the magnitude of the downward contribution from skater's body weight and body configuration, see formulas above for weight + gravity under Force gets "totaled" at foot-ground-connection. (There needs to be a further adjustment if hill-slope angle is non-zero).

Effect: If its downward magnitude is greater than body-weight, the excess goes into building future potential energy and the skater's center-of-mass (CoM) rises. If its downward magnitude is less than body-weight, there is a deduction from future potential energy as the skater's CoM falls.

Grip -- The amount of downward force is important for sideways friction to maintain "grip" -- i.e. to prevent the blade or edge or wheels from skidding out sideways perpendicular to the aiming direction.

Characteristics of downward: Contributions to vertical force are not currently propulsive, only future-propulsive. And they're based only on the current body configuration of the skater (how far the foot is out toward the side away from underneath CoM), not on accelerations or decelerations of body parts. Gravitational force is what it is regardless of which way and how fast various body parts are moving.

Managing the downward component: So there's no point in worrying about timing-coordination or quickness of up-down moves. The only thing that matters is if there's sufficient total upward-pushing work done any time during the stroke-cycle to get the average position of the total mass of the body raised up to as high off the ground as it was in the previous stroke-cycle -- otherwise the stroke-cycle sequence of motions is not repeatable.

#### longitudinal fa component

fa points longitudinally along aiming-direction a. of the skate or ski, and lies in the ground surface plane.

Formula: fa = (ffgc a) a.

Effect: Moves the skate or ski relative to the skater's center-of-mass. Large uncanceled components in this direction are a problem for control -- could result in the skate or ski jetting out from underneath the skater and dumping the skater on the ground either to the front or the rear.

No propulsive significance unless gliding friction is signficant (which we will ignore for now).

?? Further investigation:

• How are forces in this direction cancelled or controlled?
• Is there a muscle-move available which could generate substantial propulsive work, but its use is greatly restricted because of its side-effects on this fa component?

#### currently propulsive fn component

fn is propulsive in the intended pushing direction, in the ground surface plane, perpendicular ("normal") to the aiming-direction of the skate or ski

Formula: fn = (ffgc n) n

Effect: This is the main push out from the skater's body. But it's not all currently propulsive. It gets split two ways in the second conceptual stage of de-composition of forces.

### (2) split into forward-backward and sideways components

It is helpful conceptually to further de-compose fn into two components which get through the "other end" of the foot-ground interface -- definitely transmitted into the ground:

• fnx is the backward-forward component of fully-transmitted force. This component is currently propulsive. [ fnx = (fn x) x ]

• fny is the sideways component of fully-transmitted force. This component is not currently propulsive, but it does effective work to add to future propulsion by first stopping the kinetic energy toward the current pushing-direction, then creating side-weight-shift kinetic energy toward the next pushing-direction. [ fny = (fn y) y ]

It is interesting to calculate the magnitude of these components in the fully-transmitted horizontal force fn based on the corresponding components of a push-force applied at fgc from created from a muscle move at joint j.  Let

ffx = (ffgc x) x be the forward-backward component applied at fgc.

ffy = (ffgc y) y be the sideways component applied at fgc.

fx = magnitude of | ffx |

fy = magnitude of | ffy |

Then

fn = (( fx x + fy y + fz z ) n) n  =  ( fx x n + fy y n) n

and so

fnx = (fn x) x  =  (( fx x n + fy y n)) (n x) x

magnitude | fnx |  =  fx (x n)2 + fy (x n)(y n)

Since  x n = sin α  and  y n = cos α (where α is the aim-angle of the skate or ski),

?? something might be not quite right with the signs in these formulas ??

| fnx |  =  fx sin2 α + fy sin α cos α

| fnx |  =  ½ (1 cos 2α) fx + ½ (sin 2α) fy

| fny |  =  ½ (sin 2α) fx + ½ (1 + cos 2α) fy

ratios of force components applied in ffgc → transmitted + convertedfn

 aim-angle α x → x x → y y → x y → y 0° 0.00 0.00 0.00 1.00 7.5° 0.02 0.13 0.13 0.98 15° 0.07 0.25 0.25 0.93 22.5° 0.15 0.35 0.35 0.85 30° 0.25 0.43 0.43 0.75 45° 0.50 0.50 0.50 0.50 60° 0.75 0.43 0.43 0.25 90° 1.00 0.00 0.00 0.00

observations from these transmission and direction-conversion ratios:

• Magic:  Column "y → x" shows that sideways force can produce currently propulsive backward force, when transmitted through an angled skate or ski -- even if there is no forward-backward component at all in the applied force.

• Super-Magic:  even if the applied backward component fx is slightly negative for propulsion, the transmitted force can have a positively propulsive backward component, provided that the applied sideways component fy is relatively large and the aim-angle α is not too large.

• up to 45 degrees, aiming the skate or ski out further to the side improves transmission and conversion into currently propulsive backward-directed force.

## how Power is transmitted (and lost) thru the body

Power = Force * Speed = Work / Time

* main Power formula

Power - Force - Velocity trade-offs.

* Isometric force transmission.

Absorption of work (thru counter-productive counter-moves)

* Sample formulas (linear Force-Velocity, quadratic Power-Velocity)

?? [ more to be added ]

## how push-Power is converted into speed-Power

?? [ more to be added ]

## limiting factors on human propulsive performance

?? [ more to be added ]

### strategies for managing these limits

?? [ more to be added ]